PSAT/NMSQT practice test #1 section 4 question 18

This is a great one to plug in on. Say the height of the trapezoid is 2, the lower width is 5, and the upper one is 3. That way, assuming you don’t know the formula for the area of a trapezoid (which exists—see below—but which you don’t need to memorize for SAT purposes) you can break the trapezoid neatly into two right triangles and a rectangle thusly (note that my drawing is not to scale):

The rectangle has an area of 3\times 2=6, and the triangles each have an area of \dfrac{1}{2}(1)(2)=1. Total area of the trapezoid: 1+6+1=8.

Now if you do the manipulations the question asks you to do, the height gets cut in half and the bases are doubled:

The areas of the triangle are still \dfrac{1}{2}(2)(1)=1. The area of the rectangle is also unchanged: 6\times 1 = 6. Therefore, the area of the trapezoid is still 1+6+1=8.

The answer is C: the area does not change.

Since this post will be a reference point now, you can also do this using the area of a trapezoid formula:

A=\dfrac{b_1+b_2}{2}h

If you double both bases and cut the height in half, you get:

A=\dfrac{2b_1+2b_2}{2}\dfrac{h}{2}

A=\left(b_1+b_2\right)\dfrac{h}{2}

Of course, that’s equivalent to the original formula. Therefore, doubling the bases and halving the height won’t change the area of any trapezoid.

I want to stress, though, that the lesson you should take from this question is not that you need to memorize the trapezoid area formula. Rather, the lesson should be that you can 1) plug in, and 2) break more complex shapes into things like triangles and rectangles that you already know how to work with.